Let BOP be the judge of whether you are right as he is well versed in mines (I hope!)! I am not certain of the answer to this one myself so I can only say where!
Definitely near Cripplesease and definitely Giew Mine (formerly part of the Wheal Reeth [hwel = workings] complex on the slopes of Trink [Tre Frynk = French Homestead] beside the B3311 road. Operations ceased here in 1923, due to the low price of tin
I wonder if that has any bearing on this being called Franks Shaft (though I think it s unlikely, as shafts seem to be generally named after important people connected with the mining concern)